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Register a new userVirasoro central charges for Nichols algebras
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Authors
A. M. Semikhatov
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A Virasoro central charge can be associated with each Nichols algebra with diagonal braiding in a way that is invariant under the Weyl groupoid action. The central charge takes very suggestive values for some items in Heckenbergers list of rank-2 Nichols algebras. In particular, this might be viewed as an indication of the existence of reasonable logarithmic extensions of W_3==WA_2, WB_2, and WG_2 models of conformal field theory. In the W_3 case, the construction of an octuplet extended algebra---a counterpart of the triplet (1,p) algebra---is outlined.
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We establish the relationship among Nichols algebras, Nichols braided Lie algebras and Nichols Lie algebras. We prove two results: (i) Nichols algebra $mathfrak B(V)$ is finite-dimensional if and only if Nichols braided Lie algebra $mathfrak L(V)$ is finite-dimensional if there does not exist any $m$-infinity element in $mathfrak B(V)$; (ii) Nichols Lie algebra $mathfrak L^-(V)$ is infinite dimensional if $ D^-$ is infinite. We give the sufficient conditions for Nichols braided Lie algebra $mathfrak L(V)$ to be a homomorphic image of a braided Lie algebra generated by $V$ with defining relations.
It is shown that if $mathfrak B(V) $ is connected Nichols algebra of diagonal type with $dim V>1$, then $dim (mathfrak L^-(V)) = infty$ $($resp. $ dim (mathfrak L(V)) = infty $$)$ $($ resp. $ dim (mathfrak B(V)) = infty $$)$ if and only if $Delta(mathfrak B(V)) $ is an arithmetic root system and the quantum numbers (i.e. the fixed parameters) of generalized Dynkin diagrams of $V$ are of finite order. Sufficient and necessary conditions for $m$-fold adjoint action in $mathfrak B(V)$ equal to zero, viz. $overline{l}_{x_{i}}^{m}[x_{j}]^ -=0$ for $x_i,~x_jin mathfrak B(V)$, are given.
In this paper we define two Lie operations, and with that we define the bicharacter algebras, Nichols bicharacter algebras, quantum Nichols bicharacter algebras, etc. We obtain explicit bases for $mathfrak L(V)${tiny $_{R}$} and $mathfrak L(V)${tiny $_{L}$} over (i) the quantum linear space $V$ with $dim V=2$; (ii) a connected braided vector $V$ of diagonal type with $dim V=2$ and $p_{1,1}=p_{2,2}= -1$. We give the sufficient and necessary conditions for $mathfrak L(V)${tiny $_{R}$}$= mathfrak L(V)$, $mathfrak L(V)${tiny $_{L}$}$= mathfrak L(V)$, $mathfrak B(V) = Foplus mathfrak L(V)${tiny $_{R}$} and $mathfrak B(V) = Foplus mathfrak L(V)${tiny $_{L}$}, respectively. We show that if $mathfrak B(V)$ is a connected Nichols algebra of diagonal type with $dim V>1$, then $mathfrak B(V)$ is finite-dimensional if and only if $mathfrak L(V)${tiny $_{L}$} is finite-dimensional if and only if $mathfrak L(V)${tiny $_{R}$} is finite-dimensional.
In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For negative energy representations of Virasoro VOA, the same conclusion holds when $W$ is $L(0)$-graded with lowest weight greater or equal to $-3$. Relationship between the first cohomology of the VOA and that of the associated Zhus algebra is also discussed.
We formulate the generation of finite dimensional pointed Hopf algebras by group-like elements and skew-primitives in geometric terms. This is done through a more general study of connected and coconnected Hopf algebras inside a braided fusion category $mathcal{C}$. We describe such Hopf algebras as orbits for the action of a reductive group on an affine variety. We then show that the closed orbits are precisely the orbits of Nichols algebras, and that all other algebras are therefore deformations of Nichols algebras. For the case where the category $mathcal{C}$ is the category $^G_Gmathcal{YD}$ of Yetter-Drinfeld modules over a finite group $G$, this reduces the question of generation by group-like elements and skew-primitives to a geometric question about rigidity of orbits. Comparing the results of Angiono Kochetov and Mastnak, this gives a new proof for the generation of finite dimensional pointed Hopf algebras with abelian groups of group-like elements by skew-primitives and group-like elements. We show that if $V$ is a simple object in $mathcal{C}$ and $text{B}(V)$ is finite dimensional, then $text{B}(V)$ must be rigid. We also show that a non-rigid Nichols algebra can always be deformed to a pre-Nichols algebra or a post-Nichols algebra which is isomorphic to the Nichols algebra as an object of the category $mathcal{C}$.
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